Countable sets versus sets that are countable in reverse mathematics

Abstract

The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary to prove theorems of ordinary mathematics, usually working in the language of second-order arithmetic L 2 . A major theme in RM is therefore the study of structures that are countable or can be approximated by countable sets. Now, countable sets must be represented by sequences here, because the higher-order definition of ‘countable set’ involving injections/bijections to N cannot be directly expressed in L 2 . Working in Kohlenbach’s higher-order RM, we investigate various central theorems, e.g. those due to König, Ramsey, Bolzano, Weierstrass, and Borel, in their (often original) formulation involving the definition of ‘countable set’ based on injections/bijections to N. This study turns out to be closely related to the logical properties of the uncountably of R, recently developed by the author and Dag Normann. Now, ‘being countable’ can be expressed by the existence of an injection to N (Kunen) or the existence of a bijection to N (Hrbacek–Jech). The former (and not the latter) choice yields ‘explosive’ theorems, i.e. relatively weak statements that become much stronger when combined with discontinuous functionals, even up to Π 2 1 - CA 0 . Nonetheless, replacing ‘sequence’ by ‘countable set’ seriously reduces the first-order strength of these theorems, whatever the notion of ‘set’ used. Finally, we obtain ‘splittings’ involving e.g. lemmas by König and theorems from the RM zoo, showing that the latter are ‘a lot more tame’ when formulated with countable sets.